schkgg3

!>
!> SCHKGG  checks the nonsymmetric generalized eigenvalue problem
!> routines.
!>                                T          T        T
!> SGGHRD factors A and B as U H V  and U T V , where   means
!> transpose, H is hessenberg, T is triangular and U and V are
!> orthogonal.
!>                                 T          T
!> SHGEQZ factors H and T as  Q S Z  and Q P Z , where P is upper
!> triangular, S is in generalized Schur form (block upper triangular,
!> with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks
!> corresponding to complex conjugate pairs of generalized
!> eigenvalues), and Q and Z are orthogonal.  It also computes the
!> generalized eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)),
!> where alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus,
!> w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue
!> problem
!>
!>     det( A - w(j) B ) = 0
!>
!> and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
!> problem
!>
!>     det( m(j) A - B ) = 0
!>
!> STGEVC computes the matrix L of left eigenvectors and the matrix R
!> of right eigenvectors for the matrix pair ( S, P ).  In the
!> description below,  l and r are left and right eigenvectors
!> corresponding to the generalized eigenvalues (alpha,beta).
!>
!> When SCHKGG is called, a number of matrix  () and a
!> number of matrix  are specified.  For each size ()
!> and each type of matrix, one matrix will be generated and used
!> to test the nonsymmetric eigenroutines.  For each matrix, 15
!> tests will be performed.  The first twelve  should be
!> small -- O(1).  They will be compared with the threshold THRESH:
!>
!>                  T
!> (1)   | A - U H V  | / ( |A| n ulp )
!>
!>                  T
!> (2)   | B - U T V  | / ( |B| n ulp )
!>
!>               T
!> (3)   | I - UU  | / ( n ulp )
!>
!>               T
!> (4)   | I - VV  | / ( n ulp )
!>
!>                  T
!> (5)   | H - Q S Z  | / ( |H| n ulp )
!>
!>                  T
!> (6)   | T - Q P Z  | / ( |T| n ulp )
!>
!>               T
!> (7)   | I - QQ  | / ( n ulp )
!>
!>               T
!> (8)   | I - ZZ  | / ( n ulp )
!>
!> (9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
!>
!>    | l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )
!>
!> (10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
!>                           T
!>   | l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )
!>
!>       where the eigenvectors l' are the result of passing Q to
!>       STGEVC and back transforming (HOWMNY='B').
!>
!> (11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of
!>
!>       | (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )
!>
!> (12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of
!>
!>       | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )
!>
!>       where the eigenvectors r' are the result of passing Z to
!>       STGEVC and back transforming (HOWMNY='B').
!>
!> The last three test ratios will usually be small, but there is no
!> mathematical requirement that they be so.  They are therefore
!> compared with THRESH only if TSTDIF is .TRUE.
!>
!> (13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )
!>
!> (14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )
!>
!> (15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
!>            |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp
!>
!> In addition, the normalization of L and R are checked, and compared
!> with the threshold THRSHN.
!>
!> Test Matrices
!> ---- --------
!>
!> The sizes of the test matrices are specified by an array
!> NN(1:NSIZES); the value of each element NN(j) specifies one size.
!> The  are specified by a logical array DOTYPE( 1:NTYPES ); if
!> DOTYPE(j) is .TRUE., then matrix type  will be generated.
!> Currently, the list of possible types is:
!>
!> (1)  ( 0, 0 )         (a pair of zero matrices)
!>
!> (2)  ( I, 0 )         (an identity and a zero matrix)
!>
!> (3)  ( 0, I )         (an identity and a zero matrix)
!>
!> (4)  ( I, I )         (a pair of identity matrices)
!>
!>         t   t
!> (5)  ( J , J  )       (a pair of transposed Jordan blocks)
!>
!>                                     t                ( I   0  )
!> (6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
!>                                  ( 0   I  )          ( 0   J  )
!>                       and I is a k x k identity and J a (k+1)x(k+1)
!>                       Jordan block; k=(N-1)/2
!>
!> (7)  ( D, I )         where D is diag( 0, 1,..., N-1 ) (a diagonal
!>                       matrix with those diagonal entries.)
!> (8)  ( I, D )
!>
!> (9)  ( big*D, small*I ) where  is near overflow and small=1/big
!>
!> (10) ( small*D, big*I )
!>
!> (11) ( big*I, small*D )
!>
!> (12) ( small*I, big*D )
!>
!> (13) ( big*D, big*I )
!>
!> (14) ( small*D, small*I )
!>
!> (15) ( D1, D2 )        where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
!>                        D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
!>           t   t
!> (16) U ( J , J ) V     where U and V are random orthogonal matrices.
!>
!> (17) U ( T1, T2 ) V    where T1 and T2 are upper triangular matrices
!>                        with random O(1) entries above the diagonal
!>                        and diagonal entries diag(T1) =
!>                        ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
!>                        ( 0, N-3, N-4,..., 1, 0, 0 )
!>
!> (18) U ( T1, T2 ) V    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
!>                        s = machine precision.
!>
!> (19) U ( T1, T2 ) V    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
!>
!>                                                        N-5
!> (20) U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
!>
!> (21) U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
!>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
!>                        where r1,..., r(N-4) are random.
!>
!> (22) U ( big*T1, small*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (23) U ( small*T1, big*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (24) U ( small*T1, small*T2 ) V  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (25) U ( big*T1, big*T2 ) V      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
!>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
!>
!> (26) U ( T1, T2 ) V     where T1 and T2 are random upper-triangular
!>                         matrices.
!> 

NSIZES

!>          NSIZES is INTEGER
!>          The number of sizes of matrices to use.  If it is zero,
!>          SCHKGG does nothing.  It must be at least zero.
!> 

NN

!>          NN is INTEGER array, dimension (NSIZES)
!>          An array containing the sizes to be used for the matrices.
!>          Zero values will be skipped.  The values must be at least
!>          zero.
!> 

NTYPES

!>          NTYPES is INTEGER
!>          The number of elements in DOTYPE.   If it is zero, SCHKGG
!>          does nothing.  It must be at least zero.  If it is MAXTYP+1
!>          and NSIZES is 1, then an additional type, MAXTYP+1 is
!>          defined, which is to use whatever matrix is in A.  This
!>          is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
!>          DOTYPE(MAXTYP+1) is .TRUE. .
!> 

DOTYPE

!>          DOTYPE is LOGICAL array, dimension (NTYPES)
!>          If DOTYPE(j) is .TRUE., then for each size in NN a
!>          matrix of that size and of type j will be generated.
!>          If NTYPES is smaller than the maximum number of types
!>          defined (PARAMETER MAXTYP), then types NTYPES+1 through
!>          MAXTYP will not be generated.  If NTYPES is larger
!>          than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
!>          will be ignored.
!> 

ISEED

!>          ISEED is INTEGER array, dimension (4)
!>          On entry ISEED specifies the seed of the random number
!>          generator. The array elements should be between 0 and 4095;
!>          if not they will be reduced mod 4096.  Also, ISEED(4) must
!>          be odd.  The random number generator uses a linear
!>          congruential sequence limited to small integers, and so
!>          should produce machine independent random numbers. The
!>          values of ISEED are changed on exit, and can be used in the
!>          next call to SCHKGG to continue the same random number
!>          sequence.
!> 

THRESH

!>          THRESH is REAL
!>          A test will count as  if the , computed as
!>          described above, exceeds THRESH.  Note that the error is
!>          scaled to be O(1), so THRESH should be a reasonably small
!>          multiple of 1, e.g., 10 or 100.  In particular, it should
!>          not depend on the precision (single vs. double) or the size
!>          of the matrix.  It must be at least zero.
!> 

TSTDIF

!>          TSTDIF is LOGICAL
!>          Specifies whether test ratios 13-15 will be computed and
!>          compared with THRESH.
!>          = .FALSE.: Only test ratios 1-12 will be computed and tested.
!>                     Ratios 13-15 will be set to zero.
!>          = .TRUE.:  All the test ratios 1-15 will be computed and
!>                     tested.
!> 

THRSHN

!>          THRSHN is REAL
!>          Threshold for reporting eigenvector normalization error.
!>          If the normalization of any eigenvector differs from 1 by
!>          more than THRSHN*ulp, then a special error message will be
!>          printed.  (This is handled separately from the other tests,
!>          since only a compiler or programming error should cause an
!>          error message, at least if THRSHN is at least 5--10.)
!> 

NOUNIT

!>          NOUNIT is INTEGER
!>          The FORTRAN unit number for printing out error messages
!>          (e.g., if a routine returns IINFO not equal to 0.)
!> 

A

!>          A is REAL array, dimension
!>                            (LDA, max(NN))
!>          Used to hold the original A matrix.  Used as input only
!>          if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
!>          DOTYPE(MAXTYP+1)=.TRUE.
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of A, B, H, T, S1, P1, S2, and P2.
!>          It must be at least 1 and at least max( NN ).
!> 

B

!>          B is REAL array, dimension
!>                            (LDA, max(NN))
!>          Used to hold the original B matrix.  Used as input only
!>          if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
!>          DOTYPE(MAXTYP+1)=.TRUE.
!> 

H

!>          H is REAL array, dimension (LDA, max(NN))
!>          The upper Hessenberg matrix computed from A by SGGHRD.
!> 

T

!>          T is REAL array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from B by SGGHRD.
!> 

S1

!>          S1 is REAL array, dimension (LDA, max(NN))
!>          The Schur (block upper triangular) matrix computed from H by
!>          SHGEQZ when Q and Z are also computed.
!> 

S2

!>          S2 is REAL array, dimension (LDA, max(NN))
!>          The Schur (block upper triangular) matrix computed from H by
!>          SHGEQZ when Q and Z are not computed.
!> 

P1

!>          P1 is REAL array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from T by SHGEQZ
!>          when Q and Z are also computed.
!> 

P2

!>          P2 is REAL array, dimension (LDA, max(NN))
!>          The upper triangular matrix computed from T by SHGEQZ
!>          when Q and Z are not computed.
!> 

U

!>          U is REAL array, dimension (LDU, max(NN))
!>          The (left) orthogonal matrix computed by SGGHRD.
!> 

LDU

!>          LDU is INTEGER
!>          The leading dimension of U, V, Q, Z, EVECTL, and EVECTR.  It
!>          must be at least 1 and at least max( NN ).
!> 

V

!>          V is REAL array, dimension (LDU, max(NN))
!>          The (right) orthogonal matrix computed by SGGHRD.
!> 

Q

!>          Q is REAL array, dimension (LDU, max(NN))
!>          The (left) orthogonal matrix computed by SHGEQZ.
!> 

Z

!>          Z is REAL array, dimension (LDU, max(NN))
!>          The (left) orthogonal matrix computed by SHGEQZ.
!> 

ALPHR1

!>          ALPHR1 is REAL array, dimension (max(NN))
!> 

ALPHI1

!>          ALPHI1 is REAL array, dimension (max(NN))
!> 

BETA1

!>          BETA1 is REAL array, dimension (max(NN))
!>
!>          The generalized eigenvalues of (A,B) computed by SHGEQZ
!>          when Q, Z, and the full Schur matrices are computed.
!>          On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th
!>          generalized eigenvalue of the matrices in A and B.
!> 

ALPHR3

!>          ALPHR3 is REAL array, dimension (max(NN))
!> 

ALPHI3

!>          ALPHI3 is REAL array, dimension (max(NN))
!> 

BETA3

!>          BETA3 is REAL array, dimension (max(NN))
!> 

EVECTL

!>          EVECTL is REAL array, dimension (LDU, max(NN))
!>          The (block lower triangular) left eigenvector matrix for
!>          the matrices in S1 and P1.  (See STGEVC for the format.)
!> 

EVECTR

!>          EVECTR is REAL array, dimension (LDU, max(NN))
!>          The (block upper triangular) right eigenvector matrix for
!>          the matrices in S1 and P1.  (See STGEVC for the format.)
!> 

WORK

!>          WORK is REAL array, dimension (LWORK)
!> 

LWORK

!>          LWORK is INTEGER
!>          The number of entries in WORK.  This must be at least
!>          max( 2 * N**2, 6*N, 1 ), for all N=NN(j).
!> 

LLWORK

!>          LLWORK is LOGICAL array, dimension (max(NN))
!> 

RESULT

!>          RESULT is REAL array, dimension (15)
!>          The values computed by the tests described above.
!>          The values are currently limited to 1/ulp, to avoid
!>          overflow.
!> 

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!>          > 0:  A routine returned an error code.  INFO is the
!>                absolute value of the INFO value returned.
!> 

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